On Non-separable Families of Positive Homothetic Convex Bodies

Abstract

A finite family $\mathcal {B}$ of balls with respect to an arbitrary norm in $\mathbb R^d$ ($d\ge 2$) is called a non-separable family if there is no hyperplane disjoint from $\bigcup \mathcal {B}$ that strictly separates some elements of $\mathcal {B}$ from all the other elements of $\mathcal {B}$ in $\mathbb R^d$. In this paper we prove that if $\mathcal {B}$ is a non-separable family of balls of radii $r_1, r_2,\ldots , r_n$ ($n\ge 2$) with respect to an arbitrary norm in $\mathbb R^d$ ($d\ge 2$), then $\bigcup \mathcal {B}$ can be covered by a ball of radius $\sum _{i=1}^n r_i$. This was conjectured by Erdős for the Euclidean norm and was proved for that case by Goodman and Goodman (Am Math Mon 52:494–498, 1945). On the other hand, in the same paper Goodman and Goodman conjectured that their theorem extends to arbitrary non-separable finite families of positive homothetic convex bodies in $\mathbb R^d$, $d\ge 2$. Besides giving a counterexample to their conjecture, we prove that conjecture under various additional conditions.

Appendix

For the convenience of the reader we include here the Maple code used in the the proof of Theorem 2.

>eq1 := t3*x2-x1*x2+x1*y3-y2*y3; eq2 := t1*x3-x2*x3+x2*y1-y1*y3;

>eq3 := t2*x1-x1*x3+x3*y2-y1*y2;

>Eq1 := simplify(subs(y1 = 1-x1-t1, subs(y2 = 1-x2-t2, subs(y3 = 1

-x3-t3, eq1))));>Eq2 := simplify(subs(y1 = 1-x1-t1, subs(y2 = 1-x2-t2, subs(y3 = 1-x3-t3, eq2))));

>Eq3 := simplify(subs(y1 = 1-x1-t1, subs(y2 = 1-x2-t2, subs(y3 = 1-x3-t3, eq3))));

>T := solve([Eq1, Eq2], [t2, t3], explicit = true);

>F := simplify(t1+subs(T[1][1], t2)+subs(T[1][2], t3)+a*subs(T[1][1],

subs(T[1][2], Eq3)));

>F1 := factor(simplify(diff(F, t1))); F2 := factor(simplify(diff(F, x1)));

>F3 := factor(simplify(diff(F, x2))); F4 := factor(simplify(diff(F, x3)));

>F5 := factor(simplify(diff(F, a)));

>sols := solve([F1, F2, F3, F4, F5], [t1, x1, x2, x3, a], explicit =

true, allsolutions = true);